Measuring local type might not rule out single field inflation
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based on arXiv : 1104.0244. Measuring local-type might not rule out single-field inflation. Jonathan Ganc Physics Dept., Univ. of Texas July 6, 2011 PASCOS 2011, Univ. of Cambridge. Overview of presentation.

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Measuring local-type might not rule out single-field inflation.

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based on arXiv: 1104.0244

Measuring local-type might not rule out single-field inflation.

Jonathan Ganc

Physics Dept., Univ. of Texas

July 6, 2011

PASCOS 2011, Univ. of Cambridge

Overview of presentation

  • I will demonstrate that single-field, slow-roll, canonical kinetic-term inflation with a non Bunch-Davies (BD) initial state has an enhanced local-limit bispectrum

  • I will discuss the observed from this scenario in the CMB and whether it could affect the interpretation of a Planck detection of .

Local non-Gaussianity, Jonathan Ganc

Conventional wisdom

  • Single field inflation produces ≈ (5/12) (1 – ns)≈0.01, regardless of potential, kinetic term, or initial vacuum stateCreminelli & Zaldarriaga, 2004

Local non-Gaussianity, Jonathan Ganc

What is… ?

  • … the bispectrum?

the Fourier transform of the three-point function of the curvature perturbation ζ




  • … the squeezed or local limit?

when one of the wavenumbers is much smaller than the other two, e.g. k3≪k1≈k2.




  • … local-type or ?

the best-fit parameter to a target bispectrum by the family of local bispectra, i.e. those generated from a Gaussian field by .

Consider slow-roll inflation

  • Canonical action:

  • Assume slow-roll


  • Write action in terms of (Maldacena 2003):

Local non-Gaussianity, Jonathan Ganc

To quantize ζ…

  • … we promote it to an operator :The mode functions , are independent solutions of the classical equation of motion for ζ.

  • The vacuum, slow-roll mode function is

  • We can represent a non-vacuum state by performing a Bogoliubov transformation:new state has occupation number .

vacuum or Bunch-Davies state


Local non-Gaussianity, Jonathan Ganc

To calculate the bispectrum…

  • … we use the in-in formalism:

  • We find

Local non-Gaussianity, Jonathan Ganc

Bunch-Davies vs. non-Bunch-Davies


(, )


in the squeezed limit (

enhancement of in squeezed limit vs. BD!

This effect noticed only recently (Agullo & Parker 2011).

Why was it missed earlier? People expected signal only in folded limit.

What is the observable signal in the CMB from this enhancement?

  • is calculated from the CMB by fitting the observed angular bispectrum (using transfer functions and projecting onto a sphere) to that predicted by the local bispectrum

  • The angular bispectrum must be calculated numerically.

Local non-Gaussianity, Jonathan Ganc

What we find

  • If we suppose that across the wavenumbers visible today, Thus, , even for very large .

  • Such a signal is not distinguishable in the CMB.

  • However, it’s larger than predicted by the consistency relation (c.f. ).

Local non-Gaussianity, Jonathan Ganc


  • … we’ve glossed over , the phase angle between the Bogoliubov parameters and .

  • Why is this usually OK?

    • Expect to set , at early time by matching mode functions to some non-slow-roll equations.

    • Relative phase will be dominated by exponential factors in mode functions .

    • Thus, we expect . is very large, so oscillates quickly and averages out.

Local non-Gaussianity, Jonathan Ganc

What happens if ?

  • Depending on choice of , we can get large positive or negative :

Can achieve for

Local non-Gaussianity, Jonathan Ganc

What about the consistency relation?

  • The consistency relation predicts for single-field models.

  • Here we can have .Is this a counterexample?

Local non-Gaussianity, Jonathan Ganc

Is this a counterexample to the consistency relation?

  • Initial conditions are set at some time , when is inside the horizon, i.e. .

  • Non-BD terms are multiplied by. On average, the rapidly oscillating term so the above expression ; thus, we get contributions from non-BD terms.

  • In exact local limit, , and we get zero contribution from non-BD terms.

    Consistency relation does hold in exact local limit

The takeaway

  • Slow-roll single-field inflation with an excited initial state can produce an larger than expected.

  • In the more probable case, and still not detectable in the CMB.

  • If we allow the phase angle to be constant, we can get large, detectable .

    The consistency relation is a useful guideline but it holds precisely only in the exact squeezed limit.

    (A similar conclusion is reached in Ganc & Komatsu 2010).

Local non-Gaussianity, Jonathan Ganc

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